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The classical limit of polymer quantum theories yields a one parameter familyof `effective' theories labeled by \lambda. Here we consider such families forconstrained theories and pose the problem of taking the `continuum limit',\lambda -> 0. We put forward criteria for such question to be well posed, andpropose a concrete strategy based in the definition of appropriatelyconstructed Dirac observables. We analyze two models in detail, namely aconstrained oscillator and a cosmological model arising from loop quantumcosmology. For both these models we show that the program can indeed becompleted, provided one makes a particular choice of \lambda-dependent internaltime with respect to which the dynamics is described and compared. We show thatthe limiting theories exist and discuss the corresponding limit. These resultsmight shed some light in the problem of defining a renormalization groupapproach, and its associated continuum limit, for quantum constrained systems.

Effective constrained polymeric theories and their continuum limit